Introductory Functional Analysis: With Applications to Boundary Value Problems and Finite Elements

472 Pages·1998·17.14 MB·English

by  B. Daya Reddy (auth.)

27
Texts in Applied Mathematics
Editors
JE. Marsden
L. Sirovich
M. Golubitsky
W.Jäger
F. J ohn (deceased)
Advisor
G. Iooss
Springer Science+Business Media, LLC
Texts in Applied Mathematics
I. Sirovich: Introduction to Applied Mathematics.
2. Wiggins: Introduction to Applied Nonlinear Dynamical Systems and Chaos.
3. HaleIKor;ak: Dynamics and Bifurcations.
4. ChorinlMarsden: A Mathematical Introduction to Fluid Mechanics, 3rd ed.
5. HubbardlWest: Differential Equations: A Dynamical Systems Approach:
Ordinary Differential Equations.
6. Sontag: Mathematical Control Theory: Deterministic Finite Dimensional
Systems.
7. Perko: Differential Equations and Dynamical Systems, 2nd ed.
8. Seaborn: Hypergeometrie Functions and Their Applications.
9. Pipkin: A Course on Integral Equations.
10. HoppensteadtlPeskin: Mathematics in Medicine and the Life Sciences.
11. Braun: Differential Equations and Their Applications, 4th ed.
12. StoerlBulirsch: Introduction to Numerical Analysis, 2nd ed.
13. RenardylRogers: A First Graduate Course in Partial Differential Equations.
14. Banks: Growth and Diffusion Phenomena: Mathematical Frameworks and
Applications.
15. BrennerlScott: The Mathematical Theory of Finite Element Methods.
16. Van de Velde: Concurrent Scientific Computing.
17. MarsdenlRatiu: Introduction to Mechanics and Symmetry.
18. HubbardlWest: Differential Equations: A Dynamical Systems Approach:
Higher-Dimensional Systems.
19. KaplanlGlass: Understanding Nonlinear Dynamics.
20. Holmes: Introduction to Perturbation Methods.
21. CurtainlZwart: An Introduction to Infinite-Dimensional Linear Systems
Theory.
22. Thomas: Numerical Partial Differential Equations: Finite Difference
Methods.
23. Taylor: Partial Differential Equations: Basic Theory.
24. Merkin: Introduction to the Theory of Stability.
25. Naber: Topology, Geometry, and Gauge Fields: Foundations.
26. PoldermanlWillems: Introduction to Mathematical Systems Theory:
A Behavioral Approach.
27. Reddy: Introductory Functional Analysis: with Applications to Boundary
Value Problems and Finite Elements.
B. Daya Reddy
Introductory Functional
Analysis
With Applications to Boundary Value Problems
and Finite Elements
With 145 Illustrations
, Springer
B. Daya Reddy
Department of Mathematics and
Applied Mathematics
University of Cape Town
7700 Rondebosch
South Africa
Series Editors
J. E. Marsden L. Sirovich
Control and Dynamical Systems, 116-81 Division of Applied Mathematics
California Institute ofTechnology Brown University
Pasadena, CA 91125 Providencc, RI 02912
USA USA
M. Golubitsky W. Jäger
Department of M athematics Department of Applied Mathematics
University of Houston Universität Heidelberg
Houston, TX 77204-3476 Im Neuenheimer Feld 294
USA 69120 Heidelberg
Germany
Mathematics Subject Classification (1991): 46-01, 65N30
Library of Congress Cataloging-in-Publication Data
Reddy, B. Dayanand, 1953-
Introductory functional analysis: with applications to boundary
value problems and finite elements I B. Daya Reddy.
p. cm. - (Texts in applied mathematics ; 27)
Includes bibliographical re/erences and index.
I. Functional analysis. I. Title. 11. Series.
(LA320.R433 1997
5 f 5 ' . 7 - dc21 97-24052
Printed on acid-free paper.
©1998 Springer Science+Business Media New York
Originally published by Springer-Verlag New York, Inc. in 1998
Softcover reprint of the hardcover I st edition 1998
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9 8 7 6 5 432 I
ISBN 978-1-4612-6824-6 ISBN 978-1-4612-0575-3 (eBook)
DOI 10.1007/978-1-4612-0575-3 SPIN 10557902
Series Preface
Mathematics is playing an ever more important role in the physical and
biological sciences, provo king a blurring of boundaries between scientific dis
ciplines and a resurgence of interest in the modern as weil as the classical
techniques of applied mathematics. This renewal of interest, both in research
and teaching, has led to the establishment of the series: Texts in Applied Mathe
matics (TAM).
The development of new courses is a natural consequence of a .high level of
excitement on the research frontier as newer techniques, such as numerical
and symbolic computer systems, dynamical systems, and chaos, mix with and
reinforce the traditional methods of applied mathematics. Thus, the purpose
of this textbook series is to meet the current and future needs of these advances
and encourage the teaching of new courses.
TAM will publish textbooks suitable für use in advanced undergraduate and
beginning graduate courses, and will complement the Applied Mathematical
Sciences (AMS) series, which will focus on advanced textbooks and research
level monographs.
Preface
A proper understanding of the theory of boundary value problems, as op
posed to a knowledge of techniques for solving specific problems or classes
of problems, requires some background in functional analysis. The same is
true of the finite element method: there is much that can be learned and
practised - for example, the basic theory of the method, computational
aspects, and so on - without knowledge of even the most basic notions of
functional analysis. But for anyone wishing to gain a proper understanding
of qualitative aspects of boundary value problems, or of aspects of the finite
element method such as those that lead to the development of error esti
mates, some background in functional analysis is an essential prerequisite.
The issue of an adequate mathematical background is somewhat more
straightforward in the case of students of mathematics who have taken
courses in real and complex analysis, followed by a course in functional
analysis. Such students are ideally equipped to follow courses that deal with
existence theory for boundary value problems, and with qualitative aspects
of the finite element method. This text has arisen out of a recognition,
though, that there are many students, researchers, and practitioners who
have not been exposed to the kind of mathematical background j ust referred
to, but who nevertheless wish to become acquainted with the basic notions
of functional analysis and its application to the kinds of problems that arise
typically in physics and engineering.
Up to the mid-1970s the availability of source material to which such in
dividuals could refer, at least in the English language, was limited almost
entirely to the standard texts on real and functional analysis, written by
mathematicians for mathematicians having the standard background. The
viii Preface
task facing the engineer or applied scientist was thus quite daunting. For
tunately the situation has progressed markedly since then. There is now
available a wide range of texts that present functional analysis, often with
one or more applications taken from engineering and physics, in a manner
accessible to readers not having the standard prerequisites. The styles dif
fer, sometimes quite considerably, from one text to another, although this
is not a bad thing given the diversity of interests and backgrounds of the
potential readership.
This text is a furt her addition to the set of books that present func
tional analysis and its applications to nonspecialists. The approach taken
is, first, to assurne that readers have no more by way ofrelevant background
than elementary courses in linear algebra, vector analysis, and differential
equations, and wish to learn the elements of linear functional analysis.
The book begins with an introduetory ehapter, which is somewhat in
the nature of a prologue, and which presents in mostly deseriptive form
a motivation for studying functional analysis from the viewpoint of those
involved in the study of problems from physics and engineering. The re
mainder of the book is then divided into parts: Part I is devoted to linear
functional analysis, Part II to an introduction to elliptic boundary value
problems, and Part III comprises a study of the finite element method.
Two applications are treated in detail in this text: elliptic boundary
value problems and the finite element method. In both cases any prior
exposure to these areas will represent an advantage to those using this
book; indeed, it is expected that such prior exposure will in many eases
have provided the motivation to study the material presented here. The
presentation of these applieations starts more or less at the beginning,
so that those having no background in these areas could use this text to
acquire such background. On the other hand, it may be the case that the
motivation to learn functional analysis arises from an interest in an area of
application other than those treated in this text. Such readers might weil
prefer to focus on Part I of the book.
The incorporation of applications and other illustrative material is ap
proached in two distinct ways. In Part I of the book new concepts, often
of an abstract nature, are rendered more accessible by the copious use
of concrete worked examples. There is little reference in this part of the
book to applications in physics and engineering, for thc simple reason that
such examples are less weil suited to laying bare the essential features of
the many ncw concepts that accompany any introduction to functional
analysis. In Parts II and III, it is appropriate and desirable to illustrate
abstract concepts by re course to concrete problems and examples taken
from physics and engineering, and this is the approach taken here. I have
used as examples problems such as heat conduction, as weil as problems in
solid and structural mechanics - elasticity, beams, and plates - and return
regularly in Parts II and III to these examples in order to motivate and
Preface ix
illustrate aspects of the theory of elliptic boundary value problems, and
finite elements.
The style adopted in this text differs from that to be found in most texts
on analysis, in that it is adapted to the goal of making the subject matter
accessible. Thus proofs are sometimes omitted when these are feIt to shed
little additional light on the relevant topic. It will also be found in places
that the presentation of detailed mathematical argument is eschewed in
favor of a more descriptive approach, again for the purpose of rendering
the material more accessible.
In addition to the many examples, each chapter ends with a collection of
exercises for the reader. Some of these consolidate material presented in the
chapters, and many exercises serve the purpose of amplification and sup
plementation. In both cases the exercises are to be regarded as an essential
component of the text. Solutions to most of the exercises are presented at
the end of the text.
Many individuals have assisted, in various ways, in the completion of this
book. I am particularly grateful to Christiaan le Roux, Jean Lubuma, and
Sizwe Mabizela, all of whom gave most generously of their time in reading
and criticizing a preliminary version of the text. They offered detailed criti
cism on aspects of style and substance, and pointed out a number of errors.
David Davidson organized a study group which worked through most of
the book; I found the comments of this group of engineering scientists very
helpful indeed. Weimin Han deserves special thanks für his constructive
suggestions; so too does Brendt Wohlberg, who offered many suggestions
for improving the text, located errors, and also provided me with valuable
advice on the preparation of figures by computer. Shaun Courtney's expert
guidance in the mysteries of Unix, and his very willing assistance with a
variety of Jb.1EX problems, are much appreciated. Most of the figures were
prepared by Bruce Bassett and Jill Goode, while Diane Laugksch assisted
me in the typing of drafts of sections of the book. I am most grateful to
these individuals for their cheerful assistance.
I express my thanks to the staff at Springer-Verlag New York für their
expert guidance and assistance with editorial aspects, as well as their advice
on the the preparation of the manuscript using Jb.1EX.
Finally, I acknowledge with gratefulness the moral support and forbear
ance of my wife Shaada and son Jordi, who have had to spend many
evenings and weekends without my company in order that I might bring
this project to fruition.
B.D.R.
Cape Town
April 1997
Contents
Series Preface v
Preface vii
Introduction 1
I Linear Functional Analysis 21
1 Sets 23
1.1 The algebra of sets 23
1.2 Sets of numbers .. 28
1.3 IRn and its subsets 37
1.4 Relations, equivalence classes, and Zom's lemma 41
1.5 Theorem proving .... 46
1.6 Bibliographical remarks 48
1. 7 Exercises ....... . 48
2 Sets of functions and Lebesgue integration 53
2.1 Continuous functions ........... . 54
2.2 Measure of sets in lRn .......... . 61
2.3 Lebesgue integration and the space V'(O) 67
2.4 Bibliographical remarks 78
2.5 Exercises ................. . 79